Factor the given polynomial completely.
x Superscript 6 Baseline plus 16 x cubed plus 64x6+16x3+64
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box within your choice.
A.x Superscript 6 Baseline plus 16 x cubed plus 64 equals enter your response here
x Superscript 6 Baseline plus 16 x cubed plus 64x6+16x3+64equals= enter your response here
B.
The polynomial is prime.

Let's factor the given polynomial:

The expression is:

\[
x^6 + 16x^3 + 64
\]

### Step 1: Recognize the pattern
Notice that the terms \(x^6\), \(16x^3\), and \(64\) resemble a quadratic form:

\[
x^6 + 16x^3 + 64 = (x^3)^2 + 16(x^3) + 64
\]

This resembles the quadratic form \(a^2 + 2ab + b^2\), which factors into \((a + b)^2\). 

### Step 2: Factor the quadratic form
Set \(y = x^3\), so the expression becomes:

\[
y^2 + 16y + 64
\]

This is a perfect square trinomial, and it factors as:

\[
(y + 8)^2
\]

### Step 3: Substitute back \(y = x^3\)
Now, replace \(y\) with \(x^3\):

\[
(x^3 + 8)^2
\]

### Step 4: Factor \(x^3 + 8\)
\(x^3 + 8\) is a sum of cubes, which factors as:

\[
x^3 + 8 = (x + 2)(x^2 - 2x + 4)
\]

Thus, the complete factorization of \(x^6 + 16x^3 + 64\) is:

\[
(x + 2)^2(x^2 - 2x + 4)^2
\]

### Final Answer:
\[
x^6 + 16x^3 + 64 = (x + 2)^2(x^2 - 2x + 4)^2
\]

This is the fully factored form.

Would you like further details, or do you have any questions?

### Related questions:
1. How do you factor a perfect square trinomial?
2. What is the formula for factoring a sum of cubes?
3. Can all cubic expressions be factored using the sum or difference of cubes?
4. How can you recognize a quadratic form in higher-degree polynomials?
5. What other factoring techniques are useful for polynomials of degree 6?

### Tip:
When factoring polynomials, always look for common patterns such as perfect square trinomials, sums/differences of cubes, or quadratic-like structures.

Factor the given polynomial completely. x^6 + 16x^3 + 64

Learn how to completely factor the polynomial x^6 + 16x^3 + 64 using factoring techniques such as recognizing quadratic forms and the sum of cubes. This problem is suitable for students in Grades 10-12 and covers key algebraic concepts like perfect square trinomials and advanced factoring methods.

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